Hello nksunmoon,
The proof of the two equations is quite simple if you consider the charge conservation principle along with the definition of the capacitance for a classical parallel plate electrical capacitor,that is dQ=CdV.So let's work with your symbols along with those of the picture you uploaded :
From the charge conservation principle :
Qfinal_state=Qinitial_state => QC1_final+QC2_final=QC1_initial+QC2_initial =>
C1*(Vd1-V1)+C2*(Vd1-V3)=C1*(Vd2-V2)+C2*(Vd2-V3) => (eq. e2 proved)
=>C1*Vd1-C1*V1+C2*Vd1-C2*V3=C1*Vd2-C1*V2+C2*Vd2-C2*V3 =>
=>C1*Vd1-C1*V1+C2*Vd1=C1*Vd2-C1*V2+C2*Vd2=>
=>(C1+C2)*Vd1-C1*V1=(C1+C2)*Vd2-C1*V2=>
=>(C1+C2)*(Vd2-Vd1)=C1*(V2-V1)=>
=>(Vd2-Vd1)=(C1/C1+C2)*(V2-V1) (eq. e1 proved)
Regards,
Jimito13